What is Fundamental: Definition and 946 Discussions

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.

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  1. E

    Fundamental theorem of algebra

    Homework Statement There are two versions of the fundamental theorem of algebra, one that says a polynomial of degree n has n roots and the other that says a polynomial can be factored into linear and irreducible quadratic factors. Is there a quick way to see how they are the same...
  2. F

    Fundamental theorem of calculus

    Homework Statement Find the derivative of cost/t (dt) evaluated from 3 to the sqrt of x The Attempt at a Solution using the fundamental theorem of calculus, I have [cos(sqrt x)]/(sqrtx) I know I have to use the chain rule but I don't know how to go about it from here. Any tips?
  3. C

    Postdoctoral Position At Penn State In Fundamental Gravitational Theory

    Via [Ilqgs] email list: POSTDOCTORAL POSITION AT PENN STATE IN FUNDAMENTAL GRAVITATIONAL THEORY The Institute for Gravitation and the Cosmos expects two openings for post-doctoral positions starting fall 2008 in the areas of general relativity and quantum gravity. Successful...
  4. S

    Fundamental Physics reasoning books

    Hey guys...I always tend to ask a lot questions...so I was wondering if you guys know of some books out there which is very fundamental and explains its reasoning on every step of its way as if it was made for kindergarten but covers advanced topics? Fields I am interested in...
  5. I

    First fundamental theorem of calculus

    how do i interpret this geometrically F^\prime(x) = \lim_{h\to 0} \frac{F(x+h) - F(x)}{h} = f(x) that the change in area with respect to change in "width" is the "height"? i don't think that's right.
  6. N

    Fundamental Frequency of a String

    Homework Statement A string with a length of 2.5m has two adjacent resonances at frequencies 112 Hz and 140 Hz. Determine the fundamental frequency of the string? A. 14 Hz B. 28 Hz C. 42 Hz D. 56...
  7. R

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    So, here is what I have: Gravitation F = -G\frac{m_{1}m_{2}}{r^{2}} Weak Interaction ? Electromagnetism F = -\frac{q_{1}q_{2}}{4\pi \epsilon_{o}r^{2}} Strong Interaction ? I'm thinking of all of these things as actual calculable forces; but I cannot...
  8. J

    Fundamental vector projection question

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  9. S

    What is the Purpose of Life: Building a Belief System or Knowledge System?

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  10. T

    Binary numbers (fundamental) question

    I have 2 questions: FIRST I read from John H. Mathews’ Numerical Methods using MATLAB, 3E that the number 1563 can be expanded in form of 10-base expansion as: (1 × 10^{3}) − (5 × 10^{2}) + (6 × 10^{1}) + (3 × 10^{0}) …......... (1) Which results 1000 − 500 + 60 + 3 equal to 563...
  11. S

    Average Power for Frequency < Fundamental Frequency: An Analysis

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  12. N

    Newton's Laws of Motion: Fundamental Law

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  13. P

    Find Fundamental Matrix for Variable Coefficient System

    I want to find the fundamental matrix for the following system: \dotx = \left(\begin{array}{cc}1/t&0\\0&1/t\end{array}\right)x + \left(\begin{array}{rr}1&1\end{array}\right)u y = (1, 1)x There's supposed to be an x' before the first equal sign but I can't get it to work for some...
  14. E

    Fundamental Interactions Problem

    Homework Statement Imagine that I throw a ball toward the floor. It hits the floor and rebounds upward. What type of fundamental interaction between the atoms of the floor and the ball causes the ball to rebound upward? a. Strong nuclear b. Electromagnetic c. Weak nuclear...
  15. Loren Booda

    Mathematica Occurence of fundamental mathematical constants

    0, 1, phi (the Golden ratio), e, pi, delta (Feigenbaum's constant) and comparatively many other fundamental, dimensionless mathematical constants occur on the interval 0 to 5. With a potential infinity of numbers to choose from, why does such a confluence exist?
  16. V

    Find fundamental period of x(t) = cos(Pi * t)

    Homework Statement Find fundamental period of x(t)\,=\,cos\left(\pi\,t\right) Homework Equations x(t)\,=\,A\,sin\left(\omega_0\,t\,+\,\phi\right) Which has a fundamental period T\,=\,\frac{2\pi}{\omega_0} The Attempt at a Solution \omega_0\,=\,\pi <---- Right? T\,=\,\frac{2\pi}{\pi}\,=\,2...
  17. S

    Newbie's Fundamental Doubt: Wave Function & Eigen Functions

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  18. E

    Harmonics and fundamental frequency questions

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  19. M

    Power at fundamental frequency

    Hi all, reading an article I've encountered the concept of power at the fundamental frequency, google didn't help me. Could you please give to me any hint about this problem? It is at least one week I'm dealing to understand what the "fundamental power" is! there is a kind of wave, express as...
  20. B

    Fundamental identity of SR valid in GR?

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  21. J

    Fundamental things, emergent things

    Some things are clearly fundamental, and some emergent. As examples of fundamental concepts I could mention energy, momentum and location. An object has a location in space, because its' particles each have own locations. On the other hand some things are emergent, for example behaviour of a...
  22. quasar987

    Fundamental group of RP^n by recurrence?

    Fundamental group of RP^n by recurrence!? Homework Statement That's it. Find the fundamental group of RP^n by recurrence. The Attempt at a Solution It's just obvious to me that it's Z/2 no matter n but what is this recurrence argument that I'm supposed to use?
  23. P

    A few fundamental questions I'm having trouble on

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  24. quasar987

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  25. E

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  26. P

    Fundamental difference between classical and quantum mechanics

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  27. E

    List of fundamental equations and constants of the standard model

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  28. F

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  29. N

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  30. K

    Fundamental theorem of calculus

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  31. quasar987

    Is the Fundamental Group of a Pointed Space Dependent on the Base Point?

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  32. quasar987

    Topology: simple connectedness and fundamental groups

    I read on wiki* that a (pointed) topological space is simply connected iff its fundamental group is trivial. But I don't see how this in accordance with the R² caracterisation that U is simply connected iff it is path-connected and has no holes in it. Take the closed unit-disk with a point of...
  33. A

    My fundamental misunderstandings of quantum mechanics

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  34. W

    Simply Connected and Fundamental Group

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  35. K

    Fundamental problem in electrostatics

    Can someone say why it is said that in electrostatics,the test charge may be moving very slowly?Specifically,what is wrong if the test charge moves at a speed which is a significant proper fraction of speed of light?
  36. T

    Factoring operator and fundamental theorem of algebra

    I haven't taken any abstract algebra course so I do not know if this is the right section to post this question in. Anyway, I am learning differential equation right now and my prof. recently showed factorization of differential operator. For example, let D be the differential operator, he...
  37. quasar987

    Fundamental groups and homotopy type

    I know of the "result" that if two pointed spaces are homeomorphic, then the group homomorphism induced by such an homeomorphism if actually an isomorphism between the fundamental groups of these pointed spaces. But is there a link between the fundamental groups of homotopy equivalent spaces?
  38. U

    How Do I Determine the Quantity on a Graph Using Fundamental Theorem?

    http://img521.imageshack.us/img521/4549/bbav9.png The problem states that it wants the upper and lower estimate of total distance. Therefore, I used rectangles to solve for it. However, let's say I'm working on upper limits. For my initial rectangles, I use the right endpoints, but then it...
  39. T

    Fundamental Theorem of Calculus

    http://img527.imageshack.us/img527/8089/fr2rl4.gif I know part a is the fundamental theorem of calculus, but I am not quite sure how to manipulate the integral to find part i or part ii. Part b is again the fundamental theorem of calculus, but I am having a hard time solving for the...
  40. C

    Fundamental question unable to find answer

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  41. S

    Fundamental standing wave in Closed tube

    Homework Statement You blow across the open mouth of an empty test tube and produce the fundamental standing wave of the air column inside the test tube. The speed of sound in air is 344 m/s and the test tube acts as a stopped pipe. If the length of the air column in the test tube is 14.0...
  42. S

    Fundamental Frequencies of Noble Metals

    Hello, I need to know the fundamental frequencies of the noble metals "silver, gold, platinum ... etc" .. thank you very much
  43. F

    Fundamental Particle: Is It Possible?

    I have a question that may seem kind of simple, but I would like to hear your thoughts about it. Is there such a thing as a fundamental particle of which all other particles are made of, or even if is even possible for such a thing to actually exist? A particle of continuous matter (no empty...
  44. F

    Fundamental frequencies and temperature

    Homework Statement The frequency of the note f_4 is f_F. If an organ pipe is open at one end and closed at the other, what length must it have for its fundamental mode to produce this note at a temperature of T? ans= v/(4*f_F), where v is the speed of sound in air. Now the part which...
  45. P

    Fundamental Groups of the Mobius Band

    Seems to be isomorphic to Z, but I can't seem to be able to prove it. Am I right? If I am, how do I prove it?
  46. B

    Why doesn't the derivative of an integral give the value at the lower limit?

    Can someone explain a concept to me? The derivative of an integral ( whose lower limit is a real constant and whose upper limit is the variable x), is the intergrand evaluated at x as per the FTofC. I always thought about this as the limit of the integral as x approached the lower limit becuase...
  47. J

    Understanding the Fundamentals of Charge and Electric Fields

    Hello... I've just recently started studying physics for my A levels and i am finding its concepts very hard to get a handle on :frown: . One major area I am having problems with is my definitions of some of the fundamental properties of electrical circuits. For example i find charge very...
  48. P

    Fundamental matrices in inhomogenous problems?

    It seems to be the case that fundamental matrices are only used when the particular solution has an initial value of 0. Since the integral for the inhomogenous term is from (t. to t) where t. is initial value. So when t=t. the inhomogenous term must be 0. But there are cases when the particlular...
  49. S

    How Did the Understanding of Fundamental Forces Evolve Historically?

    I have a question regarding past thoughts on the fundamental forces of nature. 1) Prior to the unification of the EM force, how many fundamental forces were there thought to be? 2) When did the concept of a force first come to be? 3) How were forces described prior to this time? Where...
  50. O

    Gap in fundamental knowledge of differential equations

    This is a question where the answer is given but I don't understand how to derive it. The answer to the homogeneous eq. is: A cosh(x) + B sinh(x) with no explanation of how it is arrived at and the given answer then goes on to describe how the non-homogeneous part is solved. Diff. eq. ...
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